optimum geometry for pressure vessel attachments

7/25/2019 Optimum Geometry for Pressure Vessel Attachments

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O p t i m u m G e o m e t r y f o r P r e s s u re V e s s e l t t a ch m e n t s

Go rdo n S . B j o rkm a n , J r . 1) a nd R o w l a nd R i c ha rd s , J r . 2)

1) Structural Mechanics Consult ing, Woodstock, Vermont.

2) Floodwoods Farm, Waitsfie ld, Vermont.

A B S T R A C T

Pressure vessel a t tachments local ly st i ffen the vessel shel l and al ter the mem brane stress fie ld due to internal

pressure in the vic ini ty of the a t tachment. The problem o f determining the a t tachment shape that produces m inimum

shell s tresses due to internal pressure is s imilar to the problem in plane stress of determining the geom etry of an e last ic

inclusion that produces m inimu m stress concentrat ion in a biaxial s tress field. Ap plying a design cri terion, cal led the

harmon ic fie ld condit ion, to be sat isfied by the f 'mal vessel s tresses in the presence o f the a t tachment (inclusion) yie lds a

singular integral equation in terms of the original stress fie ld and the unkno wn geom etry of the inclusion. The result ing

solut ion is an e l l ipse o f arbi trary size where the ra t io of principal axes is inversely proport ional to the ra t io o f the

princip al strains in the original stress field. The solution not only produce s constant stress around the inclusion

bound ary and no sh ear stresses, but a lso no change in the volum e stra in energy and no elast ic rota t ions any wh ere in the

stress fie ld, which is quite remarkable . The fini te e lement method is used to i l lustra te these dramatic results . Several

com mon at tachment geom etries are evaluated and com pared to the results for the optimum ellipt ic shape.

I N T R O D U C T I O N

In the design o f pressure vessels i t is comm on pract ice to w eld a varie ty of a t tachments, such as, support pads,

l i ft ing lugs, instrument ports and re inforcements to the pressure boundary o f the vessel . These a t tachments local ly

st iffen the vessel shel l and al ter the m embra ne stress fie ld due to internal pressure in the vic ini ty o f the a t tachment.

Depen ding o n the type o f weld and g eom etry of the weld footprint , the local stress fie ld in the vessel shel l a t the

at tachment bound ary can increase significantly. While mech anical loads may create occasional high stresses a t the

at tachment/vessel intersect ion, the a t tachmen t design m ay be do minated by pressure induced stresses and their

fluctuat ions. In those si tuations w here the overal l s tress concentrat ion caused by the a t tachment m ust be l imited, i t pays

to invest igate a l ternate a t tachment geom etries that can minimize the m embran e stresses.

The problem of determining the a t tachment shape that produces minim um shell stresses due to internal pressure is

similar to the problem in plane stress of determining the geo metry o f an e last ic inclusion that produces minim um stress

concentrat ions in a biaxial s tress fie ld. Determ ining the unkno wn shape of the a t tachment bou ndary (the inclusion) to

sat isfy specific requirements impose d on the final s tress fie ld, is an inverse , or design problem in e last ic i ty.

In the structural design o f buildings, bridges and airframes, the search for optimum structural shapes has

tradit ionally been conducted u sing a stra tegy dist inct ly different from that o f standard mathematical analysis where the

governing equations are solved for a given set of boun dary condit ions on a bod y of specified geometry. Instead, some

geometric param eters are left unspecified and extra condit ions are im posed on the stress-stra in fie ld suffic ient to then

derive a configurat ion optimized to that preselected design cri terion.

Linear and shel l s tructures provide a c lassic examp le of this a l ternat ive design stra tegy in which the search fo r

optimum shapes to e l iminate bending and the resulting effects on structures has been a major historical theme. In

startl ing contrast , there is essentia l ly no cou nterpart to the design tradit ion in e lastic i ty. A 200-y ear history has

produ ced a rich and elegant mathem atical theory in both real and complex representat ion, but thus far the inverse

approach has general ly been bypassed. There are a numb er of influences one migh t postulate to explain, a t least in part,

this omission, but prob ably the m ost impo rtant factor is the lack o f c lear design cri teria for inverse e last ic@ that can be

transla ted into precise mathem atical s ta tements com parable in power and scope to the no-bend ing or minimum energy

condition in structures.

One such cri terion, the harmonic fie ld condition, has been successful ly applied to the design of holes that produce

absolute minim um stress concentrat ions [1]. The purpose of this paper is to sum marize the work that has been done to

apply the harmonic fie ld condition to the design o f inclusions that produce minim um stress concentrat ions and bring to

the a t tention of the engineering com mun ity the benefi ts that can be derived when using such shapes. Since the rigid

inclusion is a l imit ing case and approximates the beha vior o f st i ff re inforcements and w elded at tachments o n thin shel ls,

i t is se lected to i l lustra te the app licat ion of the method ology.

SMiRT 16, Washington DC, August 2001 Paper # 1603Transactions,


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H A R M O N I C F I E L D C O N D I T I O N

The p rob lem o f de te rmin ing the geom et ry o f a boun dary ( i . e ., a ho le o r inc lus ion ) invo lves des ign and no t s imply

ana lys i s. W hen the ge ome t ry o f a bound ary i s l ef t as a var iab le , one mus t p rescr ibe in s tead essen t ia l p roper t i es o f the

f ina l (d i s to r ted ) s t res s f i e ld su ff ic ien t to de te rmine a ho le o r inc lus ion shape fo r tha t des ign purpose . How ever , one

canno t a rb i t ra r i ly se lec t the loca l e f fec t o f a ho le o r inc lus ion on the o r ig ina l s t res s f i e ld s ince there may no t ex i s t a

shape tha t can p roduce tha t e f fec t . Thus the p rob lem i s p ro found ly d i f fe ren t f rom the f ir s t and second fundamen ta l

p rob lem s o f the theo ry o f e las t ic i ty , where the loca l e f fec t o f any ho le o r inc lus ion on the s t res s f i e ld i s un ique ly

de termined b y i t s kno wn g eomet ry . As such , in any inverse s t ra tegy the des ign cond i t ion to be sa t is f i ed by the f 'mal

s tress f ield is absolutely crucial .

S ince boundary cond i t ions un ique ly de te rmine the f ina l s t a te o f st res s in the fundam en ta l p rob lem s o f e las t ic i ty ,

and s ince max imu m s t resses mu s t occur on the boundary o f the ho le o r inc lus ion , i t wou ld seem tha t a des ign cond i t ion

requ i r ing , fo r examp le , tha t the s t res ses a round the boun dary be cons tan t , as i s the case o f the c i rcu la r ho le in an

i so t rop ic s tres s f ie ld , wou ld p rov ide su f f ic ien t cond i t ions fo r a so lu t ion and an op t imu m resu lt . Actua l ly , th i s approach

was succ ess fu l ly impleme n ted by Chere panov [2 ] in the des ign o f a ho le in a b iax ia l s tres s f ie ld . [Che repan ov ' s w ork

was unk now n to the au thors a t the t ime the au thors ' re search w as be ing conducted . ] The au thors in i t ia l ly cons idered the

cons tan t s t res s approach , bu t re jec ted i t, be l i ev ing i t to be the ma n i fes ta t ion o f a more p owe rfu l des ign co nd i t ion tha t

wo u ld inco rpora te the fundam en ta l p roper t i es o f the f 'mal s t res s f ie ld everywhe re , no t ju s t on the ho le bounda ry [ 1 ]. In

fac t, th is des ign requ i remen t (cons tan t bounda ry s t res s ) is ob ta ined as a spec ia l case o f the harm onic f i e ld cond i t ion

descr ibed here in wh ere bo th the s t res s f ie ld and boundary load s a re p rescr ibed as cons tan t . As such , the cons tan t

bound ary s t res s requ i rem en t i s severe ly l imi ted , and , in fac t , there i s no ho le shape tha t can sa t i s fy th i s des ign

requ i rem en t i f the o r ig ina l s t res s f i e ld i s no t cons tan t [3 ] .

I t is wel l know n tha t in an i so t rop ic s t res s f ie ld the c i rcu la r shape fo r ho les o r r ig id inc lus ions p roduces cons tan t

s t ress a round the boun dary an d min im um s t ress concen t ra t ions . Less kn ow n i s the fac t tha t the f i r s t invar ian t ( i. e . , the

sum of the no r mal s t resses , 11= ch+~2 = C~x+~y) s no t on ly c ons tan t on the boun dary , bu t i s a l so cons tan t everyw here in

the f 'mal (d i s to r ted ) s t res s f i e ld and equal in m agn i tude to the f i r s t invar ian t in the o r ig ina l (und is to r ted ) s t res s f i e ld p r io r

to the in t roduct ion o f the c i rcu la r ho le o r inc lus ion . W i th the c i rcu la r shape as a mod el , i t was dec ided to choose fo r the

des ign cond i t ion : That the f i rs t invar iant o f the st re ss or s t ra in) remain un chan ged everywhere in the f ie ld whe n the

hole or inc lus ion i s in troduced

This des ign cond i t ion has b road imp l ica t ions , s ince the f ir s t invar ian t sa ti s f ies LaP lace ' s equa t ion . Thus the

des ign cond i t ion can b e res ta ted such : T h a t t h e L a P la c ia n c o m p o n e n t o f th e s t r es s f i e l d r e m a in s u n c h a n g e d e v e ryw h e re

when the hole or inc lus ion i s in troduced . Since func t ions tha t sa t is fy LaPla ce ' s equa t ion a re ana ly t i c and a re ca l l ed

harm onic func t ions , the t e rm harm onic f i e ld cond i t ion i s u sed to descr ibe the des ign cond i t ion , and the t e rm

harm onic shapes i s u sed to describe the geomet ry o f these spec ia l ho les and inc lus ions .

S imply s ta ted : H a rm o n ic sh a p e s d o n o tp e r tu rb t h e L a P la c ia n c o m p o n e n t o f th e o r ig in a l s t re s s f i e ld . As a

consequence , harm onic shapes in t roduce on ly pu re shear (o r on ly a change in d i s to r t ion s t ra in energy) , no chang e in

vo lum e s t ra in energy , and no e las t i c ro ta tion . [ 1 , [4 ] Thus the ana logy to mem brane shapes fo r a rches and she l l s i s

comple te . Fo r such s t ruc tu res the no bend ing cond i t ion i s ac tua l ly a requ i remen t tha t there be no e last i c ro ta t ion o f any

cross sec tion , and the harm onic f i e ld cond i t ion g ives a fu l l two-d im ens iona l coun terpar t impos ing no bend ing

anyw here in the f i e ld when the harm onic shape i s in t roduced .

H A R M O N I C S H A P E S

e n e r a l S o l u t i o n

S i n ce w e s e e k t he g e o m e t ry o f a n u n k n o w n b o u n d a ry , a c o m p l e x -v a ri a b l e fo rm u l a t io n , s u c h a s p r e s e n te d b y

Musk hel i shv i l i [5 ] o ffe rs a par t i cu la r ly powerfu l ana ly t i c approach . An op en ing o f es sen t ia l ly any shape can be

descr ibed as a func t ion o f a com plex var iab le z = x + iy, and i t becom es conv en ien t to map f rom a reg ion con ta in ing a

c i rcu la r bounda ry us ing the inverse t rans fo rma t ion z = co (~) . Le t t ing ~ = pe ~, the po lar coord ina tes p , 0 m ay be

cons idered the eu rv i l inear coord ina tes o f a po in t x , y o f the z -p lane re la ted by the equ a t ion z = c0 (~) . S t resses , s t ra ins

and d i sp lacemen ts fo r the p lane p rob le m wi tho u t bod y fo rces can be expressed in t e rms o f two c omp lex func t ions ~(~)

and qJ (~), wh ich co rrespo nd to the f ina l st res s f ie ld . These com plex func t ions can , in tu rn , be rep resen ted in the fo rm

[6]:

q~(g') = ~o g') + ~ * (6 ) (1 )

~ ( ( ) = ~/ o ( ( ) + ~ * ( g )

2 )


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where 6o(~) and ~o(~) are know n functions of the original stress fie ld and ~*(~) and ~* (~) are unk now n functions that

account for the perturbat ion o f the stress fie ld due to the presents of the hole or inclusion. In the standard problem s o f

elast ic i ty the unk now n functions ~*(~) and qJ*(~) are determined from the boun dary condit ion:

o r

c o ( ~ r ) ' ( ~ r )

( c r ) + + ~ , ( c r ) = F ( c r ) ( 3 )

c o ' ( a )

x ' ~ (o ) - c o ( o ' ) ' ( o - ) _ ~ ( o ' ) = 2 / t ( g ] + ig2 = D ( c r ) ( 4 )

c o ( o )

in which cr (without a subscript) equals the value o f ~ on the bo unda ry o f the un it c irc le y (i .e . ,or = e i). Eq. 3 is for the

first fundam ental problem of e last ic i ty in which F(~) represents any self-equil ibrat ing bound ary loading, while Eq . 4 is

for the second fun damental prob lem in which g l and g2 are the know n boun dary values of the displacement compon ents

u and v imposed on y. The m ateria l propert ies ~: and 2~t ma y also be w ri t ten

for plane stress as:

and for plane stra in as:

E 3 - 0

2,u = (1 + u) x = ~ l+ v (5)

E

2u = (1 + u ) x = 3 - 4 0 (6 )

in te rms of Yo ung ' s modulus , E, and Poisson ' s ra t io o .

As previously mentioned, for standard analysis where geom etry is know n, the problem be comes the determination

of the tmkn own co mp lex functions ~* and tg*. For the inverse problem, on the other hand, the desired shape wil l be

given d i rec t ly by the unknow n m apping func t ion 0( 0) which can be found f rom the appl icable boundary condi tion only

if the unkno wn perturbat ion compo nents ~* and W* can be e l iminated. Such is the case since i t can be shown [1] that

the harmon ic fie ld condit ion sim ply requires that :

~ * ( ( ) = 0 ( 7 )

and the Cauch y-type singular integral

~ * ( o r ) do- = 0

o ' - (

( 8 )

since W*(cr) is the bo undary value o f a function holomo rphic outside ~/. Thus, m ult iplying each term in Eq.3 and Eq.4

by d~ /(~-~) an d integrat ing over the boundary, the basic equations for harmonic shapes are found to be:

o r

+ U , o = f F ( o ) do

I ~ a ~ + J ~ o ' ( , ~ ) ( , ~ - ( ) , , - . J , , -

a - ( ~ c o ( c r ) ( c r - g ' ) J c r - g J c r - g

(9)

10 )

Eq.9 an d Eq.10 determine the geo metry (co(o)) of the hole or inclusion in terms o f the know n stress fie ld (~o(O),

tVo(O)) and the forces (F(o)) or displacements (D (o)) imposed on the boundary. At fi rst glance, these expressions,


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wh ich may involve nonlinear singular integrals , are not part icularly entic ing. How ever, as demonstrated below, they

are , in fact , manageable a t least for the free fie lds appro ximated in most problems found in pract ice .

The basic equations are derived with general boundary loading an d displacement functions F(~)and D (o), an d the

significance o f this capabil ity to control the harmonic shape for part icular applications mu st be emph asized. Pract ical ly

any com bination o f internal t ract ions and displacements expressed as a function of an angular coordinate , 0, could be

exam ined as, for example, prestress with expandable l iners or shrink fi t ting.

H a r m o n i c H o l e s

Consider a biaxial s tress fie ld with principal stresses NI, N2 which h ave constant m agnitude an d direct ion at every

point in the plane such that N coincides with the x-axis and I N t l > I N 2 I . In addit ion a uniform ly distributed internal

pressure of magnitude P is a lso applied to the unknow n hole boundary. For these conditions the harmo nic hole is an

el l ipse with major axis a oriented paral le l to the direct ion of the major principal stress NI an d where the ra t io of the

majo r to m inor axes o f the e l l ipse is [ 1]:

a__: N 1 - e (11)

b N 2 P

In the absence o f internal pressure the ra t io becomes:

a N I 12)

b N 2

The result ing stress is constant on the bound ary and equal to:

c~0 = N1 + N2 (13 )

which is the absolute m inimu m stress possible for any ho le shape in a biaxial fie ld [1].

Figure 1 shows the propo rt ions o f the harmon ic hole in a biaxial fie ld where N1 = 2N2. [A numb er of a ircraft

manufacturers use these proport ions in the design o f a irplane windows.] Harm onic holes for nonconstant fie lds are

given in Reference 3. Figures 2 and 3 are plots of the principal tensi le stresses around a c ircular hole and the harmonic

hole from a f 'mite e lement solut ion fo r the ease of a biaxial s tress fie ld where N I = 6 9.0 M pa (10,000p si) and N2 = 34.5

M pa (5,000psi). Note that the stress concentration factor for the c ircular hole is 2.5, whereas for the harmo nic hole i t is

only 1.5, a significant reduetion~ k

O1 =2 0 2

0 2

_ _ t t t t t t

t t t t t t t _ _ _

pl ne str in

17

p l n e s t r e s s

a

. ~ b = a / 2

, ~ ~ O1=20 2

I I - -

~

if2

F i g u r e : H a r m o n i c H o l e in a B i a x i a l S t r e ss F i e l d


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C I R C U L A R H O L E ,

N x = 1 0 , 0 8 0 p s i , N y = 5 , 0 0 0 p s i

A N S Y S 5 . 4

M A R 2 6 2 0 01

2 0 : 4 0 : 3 8

N O D A L S O L U T I O N

S T E P = I

S U B = i

T I M E = I

Sl AVG)

P o w e r G r a p h i c s

E F A C E T = I

A V R E S = M a t

D M X = . 0 0 2 9 5 5

S M N = 4 8 1 7

S M X = 2 5 5 2 7 ~ -

4 8 1 7

7118

9 4 1 9

~ 1172

1 4 0 2 2

1 6 3 2 3

1 8 6 2 4

2 0 9 2 5

2 3 2 2 6

2 5 5 2 7

F i g u r e 2 : P r i n c i p a l T e n s i l e S t r e s s C o n t o u r s f o r a C i r c u l a r H o l e i n a B i a x i a l S t r e s s F i e l d

w h e r e N ~ = 6 9 .0 M P a 1 0 , 0 0 0 p s i ) a n d N y = 3 4 . 5 M p a 5 , 0 0 0 p s i)

A N S Y S 5 . 4

M A R 2 6 2 0 0 1

2 0 : 5 0 : 2 5


S T E P = I

S U B = I

T I M E = I

S1 AVG)


E F A C E T = I

A V R E S = M a t

D M X = . 0 0 2 9 4 5

S M N = 7 8 7 9

S M X = 1 5 3 0 9 ~ - -

Z V = i

* D I S T = I . 5 2 5

* X F = 1 . 3 5 5

* Y F = 1 . 2 3 4

Z - B U F F E R

7 8 7 9

8 7 0 5

9 5 3 0

~ 1 356

1 1 1 8 1

1 2 0 0 7

1 2 8 3 3

~ 13658

~

4484

1 5 3 0 9

H a r m o n i c E l l i p s e , a / b = 2 . 0 ; N x = i 0 , 0 0 0 p s i , N y = 5 , 0 0 0 p s i

F i g u r e 3 : P r i n c i p a l T e n s i l e S t r e ss C o n t o u r s f o r th e H a r m o n i c H o l e i n a B i a x i a l S t r e s s F i e l d

w h e r e N ~ - 6 9 . 0 M P a 1 0 , 0 0 0 p s i) a n d N y - 3 4 . 5 M p a 5 , 0 0 0 p si )


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R i g i d H a r m o n i c I n c lu s io n

A g a i n , c o n s i d e r a b i a x i a l s t r e s s f i e ld w i t h p r i n c i p a l s t r e s se s N ~ ,

N2

w h i c h h a v e c o n s t a n t m a g n i t u d e a n d d i r e c t io n a t

e v e r y p o i n t i n t h e p l a n e s u c h t h a t N 1 c o i n c id e s w i t h t h e x - a x i s a n d I N ~ I > I N 2 1 . F o r t h e s e c o n d i t i o n s [ 5]

o ( O = v o g o ; ~ o ( O = r c o (( ) ; o ( O = r co ( O

( 1 4 )

i n w h i c h

F = F = _ 1 N 1 + N 2 ) ; F ' = ~ , = __1 (N 1 - N 2 ) (15)

4 2

S u b s t i t u t i n g i n t o t h e g e n e r a l i n t e g r a l E q . 1 0 a n d c o m b i n i n g s i m i l a r t e r m s , o n e o b t a i n s

( 1 6 )

as the func t iona l equa t ion to be so lved fo r 0~ (c r ) .

L e t t i n g o ( t ~ ) h a v e t h e f o r m

al a2

o O= / + ~ + + .. .

7

( 1 7 )

t h e i n t e g r a l c a n b e e v a l u a t e d t o g i v e

- F t c - 1 ) R + + . . . + R = 0

( 1 8 )

i n w h i c h t h e a q u a n t i t i e s a re c o n s t a n t s a n d R i s a r e a l s c a l in g c o n s t a n t. E q u a t i n g s i m i l a r p o w e r s o f ~ , o n e o b t ai n s

a 1 = a 2 = . . .. . = 0; a I = ~ (19)

r x - 1 )

T h u s t h e r i g i d h a r m o n i c i n c l u s i o n i s a n e l l i p s e o f a r b i t r a r y s i z e w i t h t h e r a t i o o f m a j o r t o m i n o r a x es

a ( to - 3 ) N t + ( to + 1 ) N 2

b ( to + 1 ) N 1 + ( t o - 3 ) N 2

20)

o r , i n t e r m s o f p r i n c i p a l n o r m a l s t r a in s , s i m p l y

a__~ 2

b , ~

w h e r e a c o i n c i d e s w i t h t h e x - a x i s . I n a n i s o t r o p i c f ie l d w h e r e N~ = N2 t h e s h a p e , a s e x p e c t e d i s a c i r c le . F i g u r e 4

s h o w s t h e p r o p o r t i o n s o f t h e r i g i d h a r m o n i c i n c l u s i o n i n a b i a x i a l s t r e s s f ie l d f o r t h e c a s e o f p l a n e s t r e s s w h e r e N 1 =

2N2.

T h e s t r e s s e s o n t h e b o u n d a r y o f a n y r i g i d i n c l u s i o n a r e [ 6 ] :

( 2 1 )


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=2

t

2

~ _ _ t t t t

t t t t t t t t

t ~

q l ~

~ p

- ~ -

CY

2

F i g u r e 4 : T h e R i g i d H a r m o n i c I n c l u s i o n f o r a B i a x i a l S t re s s F i e l d

~

~

R I G I D E L L I P S E , a / b = 0 . 2 9 ; N x = I 0 , 0 0 0 p s i , N y = 5 , 0 0 0 p s i , n u = 0 . 2 5

A N S Y S 5 . 4

M A R 2 6 2 0 0 1

2 1 : 0 7 : 2 4


S T E P I

SUB =i

T I M E = I

S1 AVG)


E F A C E T = I

A V K E S = M a t

D M X = . 0 0 3 0 1 9

S M N = 7 7 8 8

S M X = i 1 9 4 3 ~ - - -

Z V = i

* D I S T = I . 3 9 2

* X F = i . 2 4 3

* Y F = 1 . 2 2 6

Z - B U F F E R

7788

825

8712

9173

9 6 3 5

~ 1 96

10558

~ 11 19

48

11943

F i g u r e 5 : P r i n c i p a l T e n s i l e S t r e s s C o n t o u r s f o r a R i g i d H a r m o n i c I n c l u s i o n i n a B i a x i a l S t r e s s F i e l d

w h e r e N x = 6 9 . 0 M P a ( 1 0 , 0 0 0 p s i ), N y = 3 4 . 5 M p a ( 5 , 0 0 0 p s i ) a n d P o i s s o n s R a t i o = 0 . 2 5


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o 'p = ( I + x > R e [ ~ , ( ) ]

o

( ~ ) J

cr 0 = (3 - x) R e[ ' (o-) J .

= (1 + ~ ) I rn[ ' (~v) ] .

p

l c o ( a )

l

(22)

How ever , b ecause of the ha rmon ic f i e ld condi t ion , ~ )* = 0

0c + D 3 - ~c)

c r p = ~ ( N I + N 2) c r o = ~ ( N 1

+ N 2 ) r p o

= 0 ( 2 3 )

4 ' 4 '

As can be s een , ha rmonic inc lus ions no t on ly g ive the requi red condi t ion

O ' p + o 0 = N 1 + N 9 ( 2 4 )

a t the in t e r face , bu t a l so no shea rs , so tha t i n fac t, t he con tac t s t re s ses a re p r inc ipa l w i th no t enden cy for s l ip to occu r a t

the in t e r face (See F igure 4) . F igures 5 and 6 a re p lo t s o f the pr inc ipa l t ens i le s t re s s and the oc tahedra l shea r s t re s s

a round the r ig id ha rm onic inc lus ion f rom a p lane s t re ss f 'mi te e l ement so lu t ion where N~ = Nx = 69 .0 Mp a (10 ,000p s i ) ,

N2 = Ny = 34 .5 M pa (5 ,000p s i ) and Poi s so n ' s ra t io equa l s 0 .25 . [The reade r should be aware o f the ve ry sma l l e r ror in

the f in i t e e l ement so lu t ion when compared to the exac t so lu t ion . ]

I t i s c l ea r tha t ha rmonic ho les mus t p roduce min imum s t re s s concent ra t ions in cons tan t f i e lds [1] , s ince maximums

and min im um s in LaP lac ian f i e lds mus t occur on the boundary . I t can a l so be show n tha t t he s ame i s t rue for a r ig id

ha rm onic inc lus ion . For exam ple , fo r p l ane s t re s s the in -p lane normal s t ra ins are g iven by ;

1 1

s o = - - - ~ ( O ' o - O O ' p ) ; ~ p = - ~ - ( c r p - O t t o ) . ( 25 )

T h u s a t t h e b o u n d a r y o f a n y r i g i d i n c l u s io n :

e o = 0 ; c ro = o c r p . (26)

and the re fore :

1 + o

a p + a 0 = ( l + o ) O ' p - c r 0 (27)

o

But the m in im um va lue pos s ib le for the f i rs t i nva r i an t i n a cons tan t f i e ld is s imply N~ + N2 so the min im um s t re s ses

poss ib le on the boundary a re :

N 1 + N 2 o

O ' p = ; o 0 = ( N 1 + N 2 ) (28)

l + v ~

wh ich a re , i n fac t , t hose for the ha rmo nic inc lus ion E q .23 . The pr oo f for p l ane s t ra in i s s imi l a r.

C O M P A R I S O N O F S T R ES S C O N C E N T R A T I O N S

To eva lua te the s ign i f i cance o f the reduc t ion in s t re s s concent ra t ion prod uced by the r ig id ha rmo nic inc lus ion ,

f in i t e e l ement so lu t ions were deve loped for o the r r ig id inc lus ion geomet r i e s inc lud ing c i rcu la r , squa re wi th sma l l comer

rad ius and rec tang ula r w i th sm a l l com er rad ius . F igures 7 , 8 and 9 a re p lo ts o f the pr inc ipa l t ens i le s t re s s a round r ig id

c i rcu la r , squa re and rec tangula r inc lus ions for a p l ane s t re s s so lu t ion where N1 - N x - 69 .0 M pa (10 ,000 ps i ) , N2 = Ny =

34 .5Mpa (5 ,000ps i ) and Poi s son ' s ra t io equa l s 0 .25 .

The re su l t s a re summ ar ized in Table 1 , and , a s can be s een , t he reduc t ion in s tre s s concent ra t ion i s s ign i f i can t. A

square or rec t angula r a t t achment , w hich co mm only o ccurs in p rac ti ce , has a s t re s s concent ra t ion fac tor o f 2 .9 wherea s

the ha rmo nic inc lus ion h as a s tre s s concent ra t ion fac tor o f on ly 1 .2 .

- . . -


9/11

T A B L E 1 . - M a x i m u m P r i n c i p a l T e n s i o n S t r es s a n d S t r es s C o n c e n t r a t i o n F a c t o r

f o r H o l e s a n d R i g i d I n c l u s i o n s ( A t t a c h m e n t s ) i n a B i a x i a l S t r e s s F ie l d

w h e r e N x - 6 9 . 0 M p a ( 1 0 , 0 0 0 p s i ) , N y = 3 4 . 5 M p a ( 5 , 0 0 0 p s i ) a n d

P o i s s o n s r a t i o = 0 . 2 5 .

Holes

Rigid

Inclusions

Hole or Inclusion

Shape

. . . . .

Harmon ic Ellipse

Exact Solution)

. . . . . .

Circular Hole

Exact Solution)

. . . . . . .

Harmon ic Ellipse

Exact Solution)

Circular

. . . . . . . . .

Square

.

Rectangular

Maximum Pr incipal

Ten sion Stress 1)

pa psi

105.6 15,309

103.4 15,000

176 25,527

172.4 25,000

. . . . .

82.3 11,943

82.7 12,000

106.6

1 9 9 . 1

197.7

15,466

28,873

28,673

Maxim um Stress

Concentration Factor

1 . 5 3

1 . 5 0

2.55

2.50

. . . . . .

1.19

1 . 2 0

1.55

2.89

2.86

O ) Based on f ini te e lement resul ts , except as noted. . . . .

Y

i

:~

R I G I D E L L I P S E , a / b = 0 . 2 9; N x = 1 0 , 0 0 0 p s i , N y = 5 , 0 0 0 p s i , n u = 0 . 2 5

ANSYS 5.4

MAR 26 2001

2 1 : 1 1 : 2 8


STEP=I

SUB =I

T I M E = I

SEQV AVG)


E F A C E T = I

A V R E S = M a t

D M X = . 0 0 3 0 1 9

SMN =7494

S M X = 1 0 7 6 5

ZV =i

*DIST=I. 392

*XF =1.243

* Y F = 1 . 2 2 6

Z - B U F F E R

7494

7858

8221

8584

8948

~ 9311

9675

~ 10038

10401

10765

F i g u r e 6 : V o n M i s e s S h e a r S t r e s s C o n t o u r s f o r a R i g i d H a r m o n i c I n c l u s i o n i n a B i a x i a l S t r e s s F i e l d

w h e r e N x - 6 9 . 0 M P a ( 1 0 , 0 0 0 p s i ), N y = 3 4 .5 M p a ( 5 , 0 0 0 p s i) a n d P o i s s o n s R a t i o = 0 .2 5


10/11

Y

L__.X

..

R I G I D C I R C U L A R I N C L U S I O N ; N X = I 0 , 0 0 0 p si , N y = 5 , 0 0 0 p s i , n u = 0 . 2 5

A N S Y S 5 . 4

M A R 2 6 2 0 0 1

2 1 : 2 4 : 5 8


S T E P = I

S U B = I

T I M E = I

Sl AVG)


E F A C E T = I

A V R E S = M a t

D M X = . 0 0 3 0 2

S M N = 6 3 7 8

S M X = 1 5 4 6 6 4 ~ - -

Z V = I

~ D I S T = I . I I 4

~ X F = . 9 7 4 4 4 5

~ Y F = . 9 5 8 9 7 7

Z - B U F F E R

6 3 7 8

~ 7388

8 3 9 8

9 4 0 7

1 0 4 1 7

[ ~ 427

1 2 4 3 7

1 3 4 4 6

14456

1 5 4 6 6

F i g u r e 7 : P r i n c i p a l T e n s i l e S t r e s s C o n t o u r s f o r a R i g i d C i r c u l a r I n c l u s i o n i n a B i a x i a l S t r e s s F i e l d

w h e r e N x - 6 9 . 0 M P a ( 1 0 , 0 0 0 p s i ) , N y = 3 4 .5 M p a ( 5 , 0 0 0 p s i) a n d P o i s s o n s R a t i o - 0 . 2 5

R I G I D S Q U A R E I N C L U S I O N ;

N x = 1 0 , 0 0 0 p s i , N y = 5 , 0 0 0 p s i , n u = 0 . 2 5

A N S Y S 5 . 4

M A R 2 7 2 0 0 1

i i : 0 3 : 0 7


S T E P = I

S U B = i

T I M E = I

S1 AVG)


E F A C E T = I

A V R E S = M a t

D M X = . 0 0 3 0 2 2

S M N = 5 9 8 8

S M X = 2 8 8 7 3 ~ - -

Z V = i

* D I S T = . 9 8 3 2 0 3

* X F = . 8 6 2 3 6 5

* Y F = . 8 4 3 4 2 9

Z - B U F F E R

5 9 8 8

6 5 0 0

8 0 0 0

i 0 0 0 0

1 2 0 0 0

1 4 0 0 0

1 6 0 0 0

2 0 0 0 0

2 8 0 0 0

F i g u r e 8 : P r i n c i p a l T e n s i l e S t r e s s C o n t o u r s f o r a R i g i d S q u a r e I n c l u s i o n i n a B i a x i a l S t r e s s F i e l d

w h e r e N x - 6 9 . 0 M P a ( 1 0 , 0 0 0 p s i) , N y - 3 4 . 5 M p a ( 5 , 0 0 0 p s i) a n d P o i s s o n s R a t i o - 0 . 2 5

I


11/11

C O N C L U S I O N

A p p l i c a t i o n o f th e h a r m o n i c f i e l d c o n d i t i o n t o t h e d e s i g n o f i n c l u s i o n s i n a b i a x i a l st r e s s f i e l d r e s u lt s i n a r i g i d i n c l u s i o n

( a t ta c h m e n t ) s h a p e i n t h e f o r m o f a n e l l i p s e w h e r e t h e r a t io o f t h e m a j o r t o m i n o r a x e s i s i n v e r s e l y p r o p o r t i o n a l t o t h e

p r in c ip a l s t ra in s in th e o r ig in a l s t res s f i e ld , o r :

C / = f 2 ( 2 9 )

b ~ ~

T h i s s i m p l y r e s u l t n o t o n l y p r o d u c e s t h e a b s o l u t e m i n i m u m s t r e s s c o n c e n t r a t i o n f o r a n y i n c l u s i o n s h a p e , b u t a l s o n o

sh ea r s t res s o n th e in c lu s io n b o u n d a ry , so th a t , in fa c t , th e co n ta c t s t res ses a re p r in c ip a l . In a d d i t io n , th e r ig id h a rm o n ic

i n c l u s i o n p r o d u c e s n o c h a n g e i n v o l u m e s t r a i n e n e r g y f r o m t h e o r i g i n a l f i e l d a n d n o e l a s t i c r o t a ti o n s a n y w h e r e i n t h e

s tress f i e ld .

R E F E R E N C E S

1 . B j o r k m a n , G . S ., J r ., a n d R i c h a r d s , R . , Jr . , H a r m o n i c H o l e s - A n I n v e r s e P r o b l e m i n E l a s t ic i t y , dou r na l o f

A pp l i ed M echani cs, A S M E , V o l . 4 3 , S e r i es E , N o . 3 , S ep t e m b e r , 1 9 7 6 , p p . 4 1 4 - 4 1 8 .

2 . C h e r e p a n o v , G . P . , I n v e r s e P r o b l e m s i n t h e P l a n e T h e o r y o f E l a s t ic i t y , Pr i k ladnaya M atemat i ka o f

M echan ika , M o s c o w , U . S . S . R . , V o l . 3 8 , N o . 6 , 1 9 7 5 , p p . 9 6 3 - 9 7 9 .

3 . B j o r k m a n , G . S . , J r ., a n d R i c h a r d s , 1 L , J r . , H a r m o n i c H o l e s f o r N o n - c o n s t a n t F i e l d s ,

d ou r n a l o f @p l i e d

M echanics, A S M E , N o . 7 8 - A P M - 3 0 , V o l . 4 6 , N o . 3 , S e p te m b e r , 1 9 7 9 , p p . 5 7 3 - 5 7 6 .

4 . R i c h a r d s , R . , J r ., a n d B j o r k m a n , G . S . , J r ., H a r m o n i c S h a p e s a n d O p t i m u m D e s i g n , dou rna l o f he

En gineer in g M echani cs D iv ision , A S C E , V o l . 1 0 6 , N o . E M 6 , D e c e m b e r , 1 9 8 0 , p p . 1 1 2 5 - 1 1 3 4 .

5 . M u s k h e l i s h v i l i , N . I . , Some B asic Prob lems in he M athemat ica l T heory of E las t ici ty , 2 nd Ed it ion , P .

N o o r d h o f f , L t d , 1 9 5 3 .

6 . S a v i n , G . N . , Stress D ist r ibu t i on A rou nd H oles, N A S A T T F - 6 0 7 , N a n k o v a D u n k a P re s s, 1 9 68 .

A N S Y S 5 . 4

M A R 27 2001

I I : 2 0 : 3 5

N O D A L

S O L U T I O N

S T E P = I

S U B

=I

T I M E = I

Sl ~VG)


E F A C E T = I

A V K E S = M a t

D M X

= . 0 0 3 0 2 8

S M N = 5 6 7 6

S M X = 2 8 6 7 3 4

ZV =i

* D I S T = I . 3 5 9

* X F = 1 . 3 0 3

* Y F = 1 . 1 0 2

Z - B U F F E R

5 6 7 6

6500

8

I0000

1 2 0 0 0

~ [ ~ 1 4

16000

20000

28000

R I G I D R E C T A N G U L A R I N C L US I O N ; N X = I 0 , 0 0 0p s i , N y = 5 , 0 0 0 p s i, n u = 0 - 2 5

F i g u r e 9 : P r i n c i p a l T e n s i l e S t r e s s C o n t o u r s f o r a R i g i d R e c t a n g u l a r I n c l u s i o n i n a B i a xi a l S t r es s F i e ld

w h e r e N x - 6 9 . 0 M P a ( 1 0 , 0 0 0 p s i ) , N y = 3 4 . 5 M p a ( 5 , 0 0 0 p s i) a n d P o i s s o n ' s R a t i o - 0 . 2 5

11

optimum geometry for pressure vessel attachments

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